Two-Level Convex Relaxed Variational Model for Multiplicative Denoising

Abstract

The fully developed speckle (multiplicative noise) naturally appears in coherent imaging systems, such as synthetic aperture radar. Since the speckle is multiplicative, it is difficult to interpret observed data. Total variation (TV) based variational models have recently been used in the removal of the speckle because of the strong edge preserving property of TV and reasonable computational cost. However, the fidelity term (or negative log-likelihood) of the original variational model [G. Aubert and J.-F. Aujol, SIAM J. Appl. Math., 68 (2008), pp. 925--946], which appears on maximum a posteriori (MAP) estimation, is not convex. Recently, the logarithmic transformation and the $m$th root transformation have been proposed to relax the nonconvexity. It is empirically observed that the $m$th root transform based variational model outperforms the log transform based variational model. However, the performance of the $m$th root transform based model critically depends on the choice of $m$. In this paper, we propose the two-level convex relaxed variational model; i.e., we relax the original variational model by using the $m$th root transformation and the concave conjugate. We also adapt the two-block nonlinear Gauss--Seidel method to solve the proposed model. The performance of the proposed model does not depend on the choice of $m$, and the model shows overall better performance than the logarithmic transformed variational model and the $m$th root transformed variational model.